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PrimeGrid
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesel Sieve
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client–server technology, BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test#LLR software, Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and FMA instruction set, Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proth Prime
A Proth number is a natural number ''N'' of the form N = k \times 2^n+1 where ''k'' and ''n'' are positive integers, ''k'' is odd and 2^n > k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. The first few Proth primes are :3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (). It is still an open question whether an infinite number of Proth primes exist. It was shown in 2022 that the reciprocal sum of Proth primes converges to a real number near 0.747392479, substantially less than the value of 1.093322456 for the reciprocal sum of Proth numbers. The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude. Definition A Proth number takes the form N=k 2^n +1 where ''k'' and ''n'' are positive integers, k is odd and 2^n>k. A Proth prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wall–Sun–Sun Prime
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Definition Let p be a prime number. When each term in the sequence of Fibonacci numbers F_n is reduced modulo p, the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted \pi(p). Since F_0 = 0, it follows that ''p'' divides F_. A prime ''p'' such that ''p''2 divides F_ is called a Wall–Sun–Sun prime. Equivalent definitions If \alpha(m) denotes the rank of apparition modulo m (i.e., \alpha(m) is the smallest positive index such that m divides F_), then a Wall–Sun–Sun prime can be equivalently defined as a prime p such that p^2 divides F_. For a prime ''p'' ≠ 2, 5, the rank of apparition \alpha(p) is known to divide p - \left(\tfrac\right), where the Legendre symbol \textstyle\left(\frac\right) has the values :\left(\frac\right) = \beg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wieferich Prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the ''abc'' conjecture. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The stronge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesel Problem
In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the form is instead k\times2^n+1, then ''k'' is a Sierpiński number. Riesel problem In 1956, Hans Riesel showed that there are an infinite number of integers ''k'' such that k\times2^n-1 is not prime for any integer ''n''. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any ''k'' less than 509203, it is conjectured to be the smallest Riesel number. To check if there are ''k'' ''k'') :2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primes In Arithmetic Progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n \le 2. According to the Green–Tao theorem, there exist arbitrarily long arithmetic progressions in the sequence of primes. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form an + b, where ''a'' and ''b'' are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For any integer k\geq 3, an AP-''k'' (also called PAP-''k'') is any sequence of k primes in arithmetic progression. An AP-k can be written as k primes of the form an+b, for fixed integers a (called the common difference) and b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesel Number
In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the form is instead k\times2^n+1, then ''k'' is a Sierpiński number. Riesel problem In 1956, Hans Riesel showed that there are an infinite number of integers ''k'' such that k\times2^n-1 is not prime for any integer ''n''. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any ''k'' less than 509203, it is conjectured to be the smallest Riesel number. To check if there are ''k'' ''k'') :2, 3, 3, 39, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 5, 7, 6, 6, 11, 7, 6, 29, 6, 6, 7, 6, 6, 7, 6, 6, 6, 8, 8, 7, 7, 10, 9, 7, 8, 9, 7, 8, 7, 7, 8, 7, 8, 10, 7, 7, 26, 9, 7, 8, 7, 7, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Germain Prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let ''p'' be a prime number of the form 8''k'' + 7 and to let ''n'' = ''p'' – 1. In this case, x^n + y^n = z^n is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime Search
Twin Prime Search (TPS) is a volunteer computing project that looks for large twin primes. It uses the programs LLR (for primality testing) and NewPGen (for sieving). It was founded on April 13, 2006, by Michael Kwok. In number theory, it is conjectured that there are infinitely many twin primes, and this is known as the twin prime conjecture. Progress TPS found a record twin prime, 2003663613 × 2195000 ± 1, on January 15, 2007, on a computer operated by Eric Vautier. It is 58,711 digits long, which made it the largest known twin prime at the time. The project worked in collaboration with PrimeGrid, which did most of the LLR tests. On August 6, 2009, those same two projects announced that a new record twin prime had been found. The primes are 65516468355 × 2333333 ± 1, and have 100,355 digits. On December 25, 2011, Timothy D Winslow found the world's largest known twin primes 3756801695685 × 2666669 ± 1. , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas–Lehmer–Riesel Test
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form with odd . The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form. For numbers of the form ( Proth numbers), either application of Proth's theorem (a Las Vegas algorithm) or one of the deterministic proofs described in Brillhart–Lehmer–Selfridge 1975 (see Pocklington primality test) are used. The algorithm The algorithm is very similar to the Lucas–Lehmer test, but with a variable starting point depending on the value of . Define a sequence for all by: : u_i = u_^2-2. Then , with , is prime if and only if it divides . Finding the starting value The starting value is determined as follows. * If : if and is even, or and is odd, then divides , and there is no need to test. Otherwise, and the Lucas sequence may be used: we take u_0 = (2+\sqrt)^k+(2-\sqrt) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berkeley Open Infrastructure For Network Computing
The Berkeley Open Infrastructure for Network Computing (BOINC, pronounced rhymes with "oink") is an open-source middleware system for volunteer computing (a type of distributed computing). Developed originally to support SETI@home, it became the platform for many other applications in areas as diverse as medicine, molecular biology, mathematics, linguistics, climatology, environmental science, and astrophysics, among others. The purpose of BOINC is to enable researchers to utilize processing resources of personal computers and other devices around the world. BOINC development began with a group based at the Space Sciences Laboratory (SSL) at the University of California, Berkeley, and led by David P. Anderson, who also led SETI@home. As a high-performance volunteer computing platform, BOINC brings together 34,236 active participants employing 136,341 active computers (hosts) worldwide, processing daily on average 20.164 PetaFLOPS (it would be the 21st largest processing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
